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G = C9×C8⋊C22order 288 = 25·32

Direct product of C9 and C8⋊C22

direct product, metabelian, nilpotent (class 3), monomial, 2-elementary

Aliases: C9×C8⋊C22, D82C18, C727C22, C36.63D4, SD161C18, M4(2)⋊1C18, C36.48C23, C8⋊(C2×C18), (C9×D8)⋊6C2, C4○D44C18, D42(C2×C18), (C2×D4)⋊5C18, Q83(C2×C18), (C3×D8).5C6, C4.14(D4×C9), C6.78(C6×D4), (D4×C18)⋊14C2, C24.12(C2×C6), (C9×SD16)⋊5C2, (C6×D4).16C6, C18.78(C2×D4), C2.15(D4×C18), C12.73(C3×D4), (C2×C18).25D4, C22.5(D4×C9), (D4×C9)⋊11C22, (C9×M4(2))⋊5C2, C4.5(C22×C18), (C3×SD16).1C6, (Q8×C9)⋊10C22, (C2×C36).67C22, C12.48(C22×C6), (C3×M4(2)).1C6, C3.(C3×C8⋊C22), (C9×C4○D4)⋊7C2, (C3×C8⋊C22).C3, (C2×C4).4(C2×C18), (C2×C6).29(C3×D4), (C2×C12).65(C2×C6), (C3×C4○D4).14C6, (C3×D4).14(C2×C6), (C3×Q8).27(C2×C6), SmallGroup(288,186)

Series: Derived Chief Lower central Upper central

C1C4 — C9×C8⋊C22
C1C2C6C12C36D4×C9C9×D8 — C9×C8⋊C22
C1C2C4 — C9×C8⋊C22
C1C18C2×C36 — C9×C8⋊C22

Generators and relations for C9×C8⋊C22
 G = < a,b,c,d | a9=b8=c2=d2=1, ab=ba, ac=ca, ad=da, cbc=b3, dbd=b5, cd=dc >

Subgroups: 174 in 102 conjugacy classes, 60 normal (36 characteristic)
C1, C2, C2 [×4], C3, C4 [×2], C4, C22, C22 [×5], C6, C6 [×4], C8 [×2], C2×C4, C2×C4, D4, D4 [×2], D4 [×2], Q8, C23, C9, C12 [×2], C12, C2×C6, C2×C6 [×5], M4(2), D8 [×2], SD16 [×2], C2×D4, C4○D4, C18, C18 [×4], C24 [×2], C2×C12, C2×C12, C3×D4, C3×D4 [×2], C3×D4 [×2], C3×Q8, C22×C6, C8⋊C22, C36 [×2], C36, C2×C18, C2×C18 [×5], C3×M4(2), C3×D8 [×2], C3×SD16 [×2], C6×D4, C3×C4○D4, C72 [×2], C2×C36, C2×C36, D4×C9, D4×C9 [×2], D4×C9 [×2], Q8×C9, C22×C18, C3×C8⋊C22, C9×M4(2), C9×D8 [×2], C9×SD16 [×2], D4×C18, C9×C4○D4, C9×C8⋊C22
Quotients: C1, C2 [×7], C3, C22 [×7], C6 [×7], D4 [×2], C23, C9, C2×C6 [×7], C2×D4, C18 [×7], C3×D4 [×2], C22×C6, C8⋊C22, C2×C18 [×7], C6×D4, D4×C9 [×2], C22×C18, C3×C8⋊C22, D4×C18, C9×C8⋊C22

Smallest permutation representation of C9×C8⋊C22
On 72 points
Generators in S72
(1 2 3 4 5 6 7 8 9)(10 11 12 13 14 15 16 17 18)(19 20 21 22 23 24 25 26 27)(28 29 30 31 32 33 34 35 36)(37 38 39 40 41 42 43 44 45)(46 47 48 49 50 51 52 53 54)(55 56 57 58 59 60 61 62 63)(64 65 66 67 68 69 70 71 72)
(1 66 59 23 43 28 53 18)(2 67 60 24 44 29 54 10)(3 68 61 25 45 30 46 11)(4 69 62 26 37 31 47 12)(5 70 63 27 38 32 48 13)(6 71 55 19 39 33 49 14)(7 72 56 20 40 34 50 15)(8 64 57 21 41 35 51 16)(9 65 58 22 42 36 52 17)
(10 29)(11 30)(12 31)(13 32)(14 33)(15 34)(16 35)(17 36)(18 28)(19 71)(20 72)(21 64)(22 65)(23 66)(24 67)(25 68)(26 69)(27 70)(46 61)(47 62)(48 63)(49 55)(50 56)(51 57)(52 58)(53 59)(54 60)
(10 24)(11 25)(12 26)(13 27)(14 19)(15 20)(16 21)(17 22)(18 23)(28 66)(29 67)(30 68)(31 69)(32 70)(33 71)(34 72)(35 64)(36 65)

G:=sub<Sym(72)| (1,2,3,4,5,6,7,8,9)(10,11,12,13,14,15,16,17,18)(19,20,21,22,23,24,25,26,27)(28,29,30,31,32,33,34,35,36)(37,38,39,40,41,42,43,44,45)(46,47,48,49,50,51,52,53,54)(55,56,57,58,59,60,61,62,63)(64,65,66,67,68,69,70,71,72), (1,66,59,23,43,28,53,18)(2,67,60,24,44,29,54,10)(3,68,61,25,45,30,46,11)(4,69,62,26,37,31,47,12)(5,70,63,27,38,32,48,13)(6,71,55,19,39,33,49,14)(7,72,56,20,40,34,50,15)(8,64,57,21,41,35,51,16)(9,65,58,22,42,36,52,17), (10,29)(11,30)(12,31)(13,32)(14,33)(15,34)(16,35)(17,36)(18,28)(19,71)(20,72)(21,64)(22,65)(23,66)(24,67)(25,68)(26,69)(27,70)(46,61)(47,62)(48,63)(49,55)(50,56)(51,57)(52,58)(53,59)(54,60), (10,24)(11,25)(12,26)(13,27)(14,19)(15,20)(16,21)(17,22)(18,23)(28,66)(29,67)(30,68)(31,69)(32,70)(33,71)(34,72)(35,64)(36,65)>;

G:=Group( (1,2,3,4,5,6,7,8,9)(10,11,12,13,14,15,16,17,18)(19,20,21,22,23,24,25,26,27)(28,29,30,31,32,33,34,35,36)(37,38,39,40,41,42,43,44,45)(46,47,48,49,50,51,52,53,54)(55,56,57,58,59,60,61,62,63)(64,65,66,67,68,69,70,71,72), (1,66,59,23,43,28,53,18)(2,67,60,24,44,29,54,10)(3,68,61,25,45,30,46,11)(4,69,62,26,37,31,47,12)(5,70,63,27,38,32,48,13)(6,71,55,19,39,33,49,14)(7,72,56,20,40,34,50,15)(8,64,57,21,41,35,51,16)(9,65,58,22,42,36,52,17), (10,29)(11,30)(12,31)(13,32)(14,33)(15,34)(16,35)(17,36)(18,28)(19,71)(20,72)(21,64)(22,65)(23,66)(24,67)(25,68)(26,69)(27,70)(46,61)(47,62)(48,63)(49,55)(50,56)(51,57)(52,58)(53,59)(54,60), (10,24)(11,25)(12,26)(13,27)(14,19)(15,20)(16,21)(17,22)(18,23)(28,66)(29,67)(30,68)(31,69)(32,70)(33,71)(34,72)(35,64)(36,65) );

G=PermutationGroup([(1,2,3,4,5,6,7,8,9),(10,11,12,13,14,15,16,17,18),(19,20,21,22,23,24,25,26,27),(28,29,30,31,32,33,34,35,36),(37,38,39,40,41,42,43,44,45),(46,47,48,49,50,51,52,53,54),(55,56,57,58,59,60,61,62,63),(64,65,66,67,68,69,70,71,72)], [(1,66,59,23,43,28,53,18),(2,67,60,24,44,29,54,10),(3,68,61,25,45,30,46,11),(4,69,62,26,37,31,47,12),(5,70,63,27,38,32,48,13),(6,71,55,19,39,33,49,14),(7,72,56,20,40,34,50,15),(8,64,57,21,41,35,51,16),(9,65,58,22,42,36,52,17)], [(10,29),(11,30),(12,31),(13,32),(14,33),(15,34),(16,35),(17,36),(18,28),(19,71),(20,72),(21,64),(22,65),(23,66),(24,67),(25,68),(26,69),(27,70),(46,61),(47,62),(48,63),(49,55),(50,56),(51,57),(52,58),(53,59),(54,60)], [(10,24),(11,25),(12,26),(13,27),(14,19),(15,20),(16,21),(17,22),(18,23),(28,66),(29,67),(30,68),(31,69),(32,70),(33,71),(34,72),(35,64),(36,65)])

99 conjugacy classes

class 1 2A2B2C2D2E3A3B4A4B4C6A6B6C6D6E···6J8A8B9A···9F12A12B12C12D12E12F18A···18F18G···18L18M···18AD24A24B24C24D36A···36L36M···36R72A···72L
order1222223344466666···6889···912121212121218···1818···1818···182424242436···3636···3672···72
size1124441122411224···4441···12222441···12···24···444442···24···44···4

99 irreducible representations

dim111111111111111111222222444
type+++++++++
imageC1C2C2C2C2C2C3C6C6C6C6C6C9C18C18C18C18C18D4D4C3×D4C3×D4D4×C9D4×C9C8⋊C22C3×C8⋊C22C9×C8⋊C22
kernelC9×C8⋊C22C9×M4(2)C9×D8C9×SD16D4×C18C9×C4○D4C3×C8⋊C22C3×M4(2)C3×D8C3×SD16C6×D4C3×C4○D4C8⋊C22M4(2)D8SD16C2×D4C4○D4C36C2×C18C12C2×C6C4C22C9C3C1
# reps11221122442266121266112266126

Matrix representation of C9×C8⋊C22 in GL6(𝔽73)

400000
040000
008000
000800
000080
000008
,
0720000
100000
00170710
001707272
00711560
00710560
,
7200000
010000
001000
0017200
00170072
00170720
,
7200000
0720000
001000
000100
00170720
00170072

G:=sub<GL(6,GF(73))| [4,0,0,0,0,0,0,4,0,0,0,0,0,0,8,0,0,0,0,0,0,8,0,0,0,0,0,0,8,0,0,0,0,0,0,8],[0,1,0,0,0,0,72,0,0,0,0,0,0,0,17,17,71,71,0,0,0,0,1,0,0,0,71,72,56,56,0,0,0,72,0,0],[72,0,0,0,0,0,0,1,0,0,0,0,0,0,1,1,17,17,0,0,0,72,0,0,0,0,0,0,0,72,0,0,0,0,72,0],[72,0,0,0,0,0,0,72,0,0,0,0,0,0,1,0,17,17,0,0,0,1,0,0,0,0,0,0,72,0,0,0,0,0,0,72] >;

C9×C8⋊C22 in GAP, Magma, Sage, TeX

C_9\times C_8\rtimes C_2^2
% in TeX

G:=Group("C9xC8:C2^2");
// GroupNames label

G:=SmallGroup(288,186);
// by ID

G=gap.SmallGroup(288,186);
# by ID

G:=PCGroup([7,-2,-2,-2,-3,-2,-3,-2,365,3110,192,5884,2951,242]);
// Polycyclic

G:=Group<a,b,c,d|a^9=b^8=c^2=d^2=1,a*b=b*a,a*c=c*a,a*d=d*a,c*b*c=b^3,d*b*d=b^5,c*d=d*c>;
// generators/relations

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